I worked for a moment in such setup (3 timezones: EU, US, Asia) in critical system with high volume of tickets.

Here are some challenges we experienced:
- if you pass the ticket to another timezone it won't necessarily speed-up the ticket resolution in case clarifications from end-user are needed (originator time-zone should be the primary contact point for the user and should make everything clear before passing tickets further)
- in our case it was worth to pass only some of critical incidents between timezones, especially when tickets were related with some global infrastructure providers which have 24h support too
- information exchange (handovers) was crucial. In our case we organized overlapping meetings between timezones, especially for hot periods. Unfortunately it resulted in 9h long working day for service managers
- KPIs definition & measurement was tricky, especially for tickets suspended on User